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Engineering Fracture Mechanics

journal homepage: www.elsevier.com/locate/engfracmech

Critical comparison of the boundary effect model with cohesivecrack model and size effect law

Christian Carlonia, Gianluca Cusatisb, Marco Salviatoc,⁎, Jia-Liang Led,Christian G. Hoovere, Zdeněk P. Bažantfa Chair, ACI Committee 446, Fracture Mechanics; and Associate Professor, Civil and Environmental Engineering, Case Western Reserve University,Cleveland, Ohio, 44106, USAb Past Chair, ACI Committee 446, Fracture Mechanics; and Associate Professor, Civil and Environmental Engineering, Northwestern University,Evanston, Illinois, 60208, USAcAssistant Professor, William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195-2400, USAdAssociate Professor, Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, Minnesota, 55455, USAeAssistant Professor, School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, Arizona, 85281, USAfMcCormick Institute Professor and W.P. Murphy Professor of Civil and Mechanical Engineering and Materials Science, Northwestern University,Evanston, Illinois, 60208, USA

A R T I C L E I N F O

Keywords:Quasibrittle materialsConcrete structuresFracture mechanicsEnergetic size effectSize effect lawStatistical size effectSize effect justificationAnalysis of experimental dataDesign codes

A B S T R A C T

For several decades it has been clear that the size effect on structural strength, exhibiting a majornon-statistical component, is a quintessential property of all quasibrittle materials. However,progress in design codes and practice for these materials has been retarded by protracted con-troversies about the proper mathematical form and justification of the size effect law (SEL). Afresh exception is the American Concrete Institute which, in 2019, becomes the first concretecode-making society to adopt the SEL based on quasibrittle fracture mechanics. This article be-gins by discussing several long-running controversies that have recently abated, and then focusescritically on the so-called Boundary Effect Model (BEM), promoted for concrete relentlessly fortwo decades, in ever-changing versions, by Xiaozhi Hu et al. The BEM is here compared to thequasibrittle SEL based on asymptotic matching. Its errors, weaknesses and inconsistencies areidentified—including incorrect large- and small-size asymptotic size effects, conflicts with broad-range comprehensive test data and with the cohesive crack model, incorrect aggregate-size de-pendence of strength, illogical dependence on ligament stress profile, inability to capture thestatistical part of size effect at large sizes, simplistic effect of boundary proximity, and lack ofdistinction between Type 1 and 2 size effects. In contrast to the SEL, the BEM is not applicable tomixed and shear fracture modes and to complex geometries of engineering structures, and is nottransplantable from concrete to other quasibrittle materials.

The purpose of this critique is to help crystallize a consensus about the proper size effectformulation, not only for concrete structures but also, and mainly, for other quasibrittle materialsand structures, including airframes made of fiber composites, ceramic components and micro-meter-scale devices, and for failure assessments of sea ice, rock, stiff soils, bone, and various bio-or bio-mimetic materials, for all of which the non-statistical size effect is yet to be widely ac-cepted in practice.

https://doi.org/10.1016/j.engfracmech.2019.04.036Received 1 February 2019; Received in revised form 15 April 2019; Accepted 28 April 2019

⁎ Corresponding author.E-mail address: salviato@aa.washington.edu (M. Salviato).

Engineering Fracture Mechanics 215 (2019) 193–210

Available online 08 May 20190013-7944/ © 2019 Elsevier Ltd. All rights reserved.

T

1. Introduction to problem faced

The scaling is a fundamental property of every physical theory—if it is not understood, the theory itself is not understood. Forelastic and elasto-plastic structures, this dictum has required no attention because their scaling law is trivial—the size effect onstructural strength is nil (except for Weibull statistical size effect, which is not discussed here). However, the situation changed withthe emergence in 1984 of the energetic size effect law (SEL) for quasibrittle failure [1,2], for which correct size effect predictionsnecessitate the theories of quasibrittle or cohesive fracture, and of damage mechanics. For such theories, exact analytical solutions arenot feasible.

So the objective has been to find an accurate enough model capturing the transition between the simple power laws that char-acterize the opposite asymptotic behaviors. This has been a challenge that has led to extensive polemics and, unfortunately, persistentcontroversies. One controversy, which has been particularly persistent, is the subject of this comparative study.

2. Progress-retarding controversies to be overcome

As a result of experiments begun in the 1970s, the importance of size effect for concrete and other quasibrittle materials forstructural design codes has been recognized [3–8]. The introduction in the 1980s of Weibull statistical power-law size effect [9–12]into the shear specifications of the design code of the Japan Society of Civil Engineers (JSCE) [13] was a pioneering act for thosetimes. Later, though, it transpired that the Weibull’s statistical power-law size effect, still embedded in the JSCE specifications [13], isnot correct for their intended purpose since a dominant large crack grows stably before failure at a specific location dictated bymechanics.

In the American Concrete Institute (ACI), design code changes depend on democratic voting of large committees dominated bycautious practitioners, who insist on broad verification and broad consensus. Inevitably, change is slow and requires protracteddiscussions. After discussions lasting for more than three decades, and thanks partly to the efforts of ACI Committee 446, FractureMechanics (chaired currently by the first author, C.C., and previously by the second author, G.C.), which approved the SE in 2007(unanimously, under K. Gerstle chairmanship) [90], a consensus has finally crystallized in the ACI. The result is that the SEL has beenapproved for the shear of beams and punching shear of slabs, and for the compression struts of the strut-and-tie model and is nowincorporated into the 2019 update of the ACI design code, ACI Standard 318. The cautiousness saved ACI from enacting anythingwrong. The ACI thus becomes the first code-making society that implicitly accepts the quasibrittle fracture mechanics as a basis of thecode.

The long delay has largely been caused by persistent controversies about alternative theories of size effect. An early controversialidea, advanced in 1993, claimed that the size effect in concrete fracture was caused by the fractal nature of concrete microstructureand its fracture surfaces. This controversy has abated after the critique, discussion and rebuttal in [14,15], and seems no longer tohinder progress in design codes.

The European fibModel Code 2010 [16] has also included a size effect for shear failures in their specifications. But its derivation issimplistic and, from the mechanics viewpoint, not justified. It is based on a controversial notion, surviving for almost three decades[18,22,23,26], namely that the size effect in shear failures of reinforced concrete beams is somehow governed by the longitudinalstrain [23] or by the width of the dominant shear crack. This latter crack itself is imagined to be governed by the longitudinal strainor by spacing of parallel shear cracks as they form at about one half of the failure load [22]. This hypothetical empirical approach hasnot received traction in ACI. But it made its way into the current European concrete design code [16]. A variant [17–29] is nowproposed for the 2020 European concrete design code (Model Code 2020), but faces serious criticism [30].

This proposed Model Code version was modified so as to give a size effect curve similar to SEL [23]. It estimates the crack widthfrom the longitudinal strain of concrete calculated for a certain point by the beam bending theory with plane cross sections remainingplane [30, c.f.]. However, 14% differences from the SEL exist. The derivation, based on beam mechanics with plane cross sections, issimplistic, illogical and misleading to students trying to understand the cause of size effect. In ACI, though, this approach received notraction.

On the other hand, another old controversy, initiated in 2000 by an article of Hu and Wittmann [31], still goes on unabated andgenerates doubts. In this article, Hu and Wittmann proposed to explain and model the size effect by the so-called Boundary EffectModel (BEM), not to be confused with the Boundary Element Method used for the solution of boundary value problems in mechanics[32, e.g]. Subsequently, various versions of BEM and critiques of the competing model were iterated in about twenty papers byXiaozhi Hu et al. [33–36, e.g.], culminating in 2017 with [37]. In response to previous criticisms [39,40], Hu et al. modified theirBEM empirically again and again, but without abandoning its faulty original premise (to be explained here) [31,33–37]. Hu et al.repeatedly claim [31,33–37, e.g.] that the size effect in reinforced concrete is caused by a boundary layer consisting of aggregateswhose finite size causes different properties. Such a layer, of course, exists, but is automatically taken into account through thefiniteness of the representative volume element (RVE) of the material and of the fracture process zone (FPZ). But the main size effectsource in cohesive fracture mechanics is the release of stored energy during stable growth of a crack with a finite fracture processzone (FPZ), and the associated stress redistribution, which is ignored by the BEM.

In their last critique of SEL, Hu et al. [37] cited and used the comprehensive concrete fracture experiments of Hoover et al.[41,42] at Northwestern University, which explored a very broad size range (1: 12.5) and notch lengths (0 to 30% of cross section),and had an exceptionally low coefficient of variation of regression errors (only 2%), thanks partly to casting all the specimens fromone and the same batch of concrete.

However, Hu et al. [37] did not cite and use Hoover et al.’s subsequent analysis of these tests in [43]. This analysis showed

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excellent agreement with the SEL and with cohesive fracture mechanics in general, and allowed calibrating the shape of the cohesivesoftening stress-displacement relation with a size-independent fracture energy. In fact, the motivation for Hoover et al.’s meticulouslyconducted tests was to disprove the hypotheses of the BEM, as explained in what follows. Hu et al. further ignored the subsequentarticle [38] which demonstrated an excellent agreement of an asymptotic-matching formulation with Hoover et al.’s test data.

By contrast, Hu et al. [37] did cite the subsequent tests by Siddık Şener et al. [44] at Gazi University in Ankara. These were near-duplicates of Hoover et al.’s tests. But they were smaller in scope and exhibited such a high scatter that any differences among the sizeeffect theories could hardly be discerned. The scatter of the data in Şener’s article [44] is, in fact, so high that it prevents disprovingBEM.

Although the BEM controversy eventually failed to prevent the adoption of the SEL by the ACI, it doubtless still rests on the mindof some members of concrete code committees in Europe and Asia, as well as all the designers of large concrete structures afflicted bysize effect. Any whiff of controversy tends to retard modernization of the design practice and codes. Thus, it is important to clarify thedeficiencies of BEM by rigorous mathematical arguments based on fracture mechanics, which is the objective of this article.

In contrast to the American concrete research community, in which the size effect law based on cohesive fracture mechanics hasby now become a consensus, the worldwide research and engineering communities dealing with other quasibrittle materials have notyet accepted the energetic size effect of cohesive fracture mechanics. They still regard the Weibull statistics [10–12] as the onlysource of size effect, even though this effect is, in reinforced concrete shear failures, either nonexistent or only a negligible con-tributor.

In particular, this is the situation for other quasibrittle materials including polymer-fiber composites, tough or toughened cera-mics, sea ice, many rocks, landslides, bio-materials (e.g., bone), nano-composites, polysilicon and micrometer scale devices[1,2,12,40,45–51,53–73]. For structures made of these materials, the size range from lab specimens to engineering applications oftenspans several orders of magnitude. As a notable example, one can think of the scaling of structural strength from composite couponsfeaturing gauge lengths of few hundreds of millimeters to the composite wing of an aircraft such as the Boeing 787 characterized by aspan of many meters. Another example is the scaling of structural strength from lab specimens of few millimeters to microdevicessuch as MEMS or electronic circuits featuring characteristic lengths on the order of microns.

Young researchers entering these fields are inevitably getting confused by seeing competing theories, such as the BEM. For them,rigorous and experimentally supported explanation has thus become critically important. The size effect problem was tackled first forconcrete because many civil engineering structures are far larger than any feasible tests, and because the fracture process zone length,about 1 ft., manifests itself clearly in normal laboratory tests. However, in view of the aforementioned dictum, the size effect is, asidefrom concrete, equally important for all quasibrittle materials. Casting doubt on the size effect theory for concrete confuses engineersdealing with the size effect in any of these materials.

3. Review of Type 1 and 2 size effect laws: Theoretical Background

Due to the complex mesostructure, the fracturing process of quasibrittle materials (such as concrete, ceramics, rocks, rigid foams,fiber composites, wood sea ice, stiff soils, consolidated snow and many bio-materials) generally features a significant stress redis-tribution. This phenomenon, which is typically characterized by the emergence of a non-linear fracture process zone (FPZ), leads to adeterministic size effect in structures weakened, prior to maximum load, by stably grown large cracks, or by notches. This type of sizeeffect is generally referred to as Type 2 size effect [1,2,45–51].

In the case of quasibrittle structures of positive geometry [50], failing at crack initiation from a smooth surface, the size effect iscaused by a combination of stress redistribution in the FPZ with strength randomness. In fact, because of the material heterogeneity, afinite FPZ must develop before the cracking can coalesce into a macrocrack of finite length emanating from the surface. At the sametime, the location of crack initiation will depend on the spatial random distribution of material strength. This type of energetic-statistical size effect is generally referred to as Type 1 size effect [12,51,53,54,56,57,60,62–64,74].

The following sections aim at providing a brief overview of Type 1 and 2 size effect laws and their physical foundations. Theinterested reader can find a more comprehensive description in the abundant literature produced on this topic in the past fourdecades [1,2,12,40,45–71,73].

3.1. Size effect in structures featuring large cracks or notches (Type 2 size effect)

In quasibrittle structures, the size of the non-linear FPZ occurring in the presence of a large stress-free crack is usually notnegligible [1,46–48,50,65,69–71,73]. The stress field along the FPZ is nonuniform and decreases with crack opening, due to dis-continuous cracking, crack deflection, and frictional pullout of grains or inhomogeneities. In this scenario, the theories such as LinearElastic Fracture Mechanics (LEFM) fail to capture the Type 2 size effect, since they inherently assume the FPZ to be negligible andhence lack a characteristic length representing the fracturing process.

This deficiency can be overcome by equipping LEFM with a fracture length scale by means of the simple approach of equivalentlinear elastic fracture mechanics (LEFM). According to this approach, the elastic crack is extended by distance r1 so that the resultantof the stress field induced by the equivalent crack (area EBCD in Fig. 1) matches the resultant of the cohesive stresses in the FPZ (areaOBCO in Fig. 1). According to the Saint-Venant principle, the stress field sufficiently far from the tip is equivalent to the one inducedby the FPZ. Accordingly, the equivalent approach provides a good approximation of the nonlinear behavior of the FPZ provided thatthe structure is sufficiently large compared to the crack size.

The size effect in geometrically similar structures is generally characterized by the effect of structure size D on the nominal

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strength =σ c P A/N s which is a measure of the load P with a dimension of stress; cs = conveniently chosen dimensionless constant,used to make σN equal to the stress at a certain chosen point (for beam specimens, we choose =c L D3 /2s where L = beam length); A= characteristic cross section area. For two-dimensional scaling, =A bD where b= structure width, and D= characteristic structuresize. Both D and A must be measured at homologous locations. For the case of elastic and elastoplastic behaviors, σN is constant,which is considered as the case of no size effect.

Let us consider a crack of effective length = +a a cf0 , where =a0 initial crack length and = =c rf 1 effective FPZ length, propor-tional to Irwin’s characteristic length =l EG f/ch f t

2 [75] and regarded as a material property [61]. The initial relative crack length=α a D/0 0 at maximum load is assumed to be a constant, which is usually true for structures of different sizes and in fracture

specimens is enforced by cutting similar notches. Following LEFM, the energy release rate can be written as follows [50]:

=′

G ασ D

Eg α( ) ( )N

2

(1)

where = =α a D/ normalized effective crack length; ′ = =E E Young’s modulus for plane stress and ′ = −E E ν/(1 )2 for plane strainwith =ν Poisson’s ratio; and =g α( ) dimensionless energy release function. Note that g α( ) accounts for geometry effects and can becalculated by finite element analysis (FEA) or using the fracture mechanics handbooks (see e.g. [76,77]). We may consider that σNattains its peak value, referred to as the nominal strength σNc of the structure, when the energy release rate at the equivalent crack tipreaches a critical value. Therefore, we have

+ =′

+ =G α c Dσ D

Eg α c D G( / ) ( / )f

Ncf f0

2

0 (2)

where Gf is the initial mode I fracture energy of the material (i.e., the area under the initial tangent of the softening curve σ w( ) ofcohesive stress σ versus crack opening w [39,61] as shown in Fig. 2) and cf is the effective FPZ length, both assumed to be materialproperties. Note that this equation characterizes the peak load if ′ >g α( ) 0, which is called the case of positive geometry [50].

By approximating g α( ) with its Taylor series expansion at =α α0 and retaining only the first two terms, one obtains:

=′

⎛⎝

+ ′ ⎞⎠

Gσ D

Eg α

cD

g α( ) ( )fNc f2

0 0 (3)

Fig. 1. Equivalent crack for a progressively softening material with cohesive stresses described by a parabolic distribution of degree n.

Fig. 2. Bi-linear cohesive law featuring the initial fracture energy, Gf , and the total fracture energy, GF .

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which can be rearranged in the following form [50]:

=′

+ ′σ

E GDg α c g α( ) ( )Nc

f

f0 0 (4)

where ′ =g α g α α( ) d ( )/d0 0 .This equation relates the nominal strength of geometrically scaled structures to a representative size, D. It can be rewritten in the

following form:

=+

σBf

D D1 /Nc

t

0 (5)

where = ′ ′Bf E G c g α/ ( )t f f 0 and = ′D c g α g α( )/ ( )f0 0 0 are constants, depending on both FPZ size and specimen geometry ( ft = tensilestrength, introduced to make B dimensionless). Thanks to the functions g α( ) and ′g α( ), Eqs. (4) and (5) explicitly account forgeometry effects and can be readily applied to any structure geometry and relative crack length. Contrary to the classical theory ofLEFM, Eq. (5) is endowed with a characteristic length scale D0. This is the key to describe the transition from quasi-ductile to brittlebehavior with increasing structure size, as seen in all quasibrittle structures. As can be noted from Fig. (3), which shows the fitting ofsize effect data of several quasibrittle materials, the agreement between Type 2 SEL and experiments is excellent.

It may be emphasized that Eqs. (4) and (5) are rigorously derived from fracture mechanics by introducing an effective FPZ lengthand using Taylor series expansion of the dimensionless energy release rate function. No empirical argument has been made. Fur-thermore, it has been shown that Eqs. (4) and (5) can be derived by leveraging other rigorous mathematical approaches, including thedimensional analysis and asymptotic matching [51] (see the Appendix section in the present manuscript) and cohesive zone modeling[61]. Eq. (5), without Eq. (4), can also be derived from dimensional analysis and the conditions of energy conservation during crackgrowth, without any recourse to fracture mechanics; see the Appendix.

It is worth noting that Eqs. (4) and (5) depend on only two fracture material parameters: the initial fracture energy Gf and theeffective FPZ size cf . The elastic modulus ′E can be characterized from uniaxial tests whereas g α( ) and ′g α( ) can be calculated for anygeometry and crack length by finite element analysis. Furthermore, cohesive zone modeling by Cusatis and Schauffert [61] showedthat an explicit relation exists between the effective FPZ length cf and Irwin’s characteristic length = ∗l E G f/ch f t

2. They showed that,for a linear cohesive law, ≈ = ∗c l E G f0.44 0.44 /f ch f t

2. This illuminating relation indicates that Eqs. (4) and (5) can be used not only toestimate the initial fracture energy but also the strength of the material. More importantly, Cusatis and Schauffert’s results prove,unequivocally, that the Type 2 Size Effect Law (SEL), Eqs. (4) and (5), can be derived as the large size asymptote of the cohesive zonemodel. This is another confirmation of the validity of SEL.

3.2. Size effect for Crack Initiation (Type 1 size effect)

The Type 1 size effect applies to structures for which there is no notch, i.e. =a 00 . In this case, the size effect is induced by bothstress redistribution in the emerging FPZ and the randomness of material strength. The effects of the FPZ can be captured by an

Fig. 3. Optimum fitting of size effect data by using Eq. (5).

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equivalent LEFM approach similar to the one used to derive the Type 2 size effect. Since, for specimens with no notches, the first termof the dimensionless energy release rate function is zero ( = =g g(0) 00 ), one can truncate the Taylor series expansion of g α( ) onlyafter the quadratic term. Then, substituting into Eq. (2) and rearranging, one gets [78]:

=′ + ″

∗σ

E G

c g g c D(0) (0) /Nc

f

f f12

2(6)

Eq. (6) may be approximated as

= −∞−σ f x(1 )Nc r

1/2 (7)

=′

=− ″

′∞

∗f

E Gg c

xg

gcD

where ,2r

f

f

f

0

0

0 (8)

Because only the first two terms of the expansion matter, Eq. (7) can further be equivalently replaced by any binomial expansion thatretains the first two terms. Therefore, it is legitimate to replace this equation by

= + = +∞ ∞σ f x σ f r x(1 ) or [1 ( /2)]Nc r Nc rr1/2 1/ (9)

where r is an arbitrary positive number determined by optimum fitting of the experimentally measured size effect curve (themodification by Eq. 9 is not only legitimate but also necessary to avoid wrong small-size asymptotics). Upon setting =x D D/2 /b

where

=− ″

′D

gg

c4b f

0

0 (10)

one gets the final form of the Type 1 size effect law (Fig. 4a):

= ⎛⎝

+ ⎞⎠∞σ f rD

D1Nc r

br1/

(11)

where ⟨ ⟩x , defined as xmax( , 0), is used in Eq. (10) because the ratio ″ ′g g/0 0 can sometimes be positive, in which case there can be nosize effect, i.e., Db must vanish.

It should be pointed out that Eq. (11) not only has the correct first and second-order asymptotic properties for → ∞D , like Eq. (7),but also has a realistic form for small D, such that → σlimD N0 be neither 0 nor imaginary. The fact that the limit → ∞σN is shared by thefamous, widely used, Petch-Hall formula for the effect of crystal size in polycrystalline materials [79] is incidental. In fact, based onthe cohesive crack model, the limit for σNc is finite, but this does not matter in practice because D cannot be less than about threemaximum aggregate sizes, as the material could no longer be treated as a continuum.

The foregoing derivation of Type 1 size effect is anchored solely by the energetic analysis of the fracture process. It predicts avanishing size effect at the large size limit. For this reason Eq. (11) is often called the energetic since it accounts for the energetic sizeeffect induced by the stress redistribution in the FPZ but not for the one produced by the randomness of material strength. However,

Fig. 4. Type 1 size effect: (a) schematic of energetic and energetic-statistical Type 1 size effect curves, and (b) comparison with experimental data.

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for very large size structures, the size effect of this type tends to be statistical, which is due to fact that the Type 1 failure can initiateat many different locations within the material, whose strength is a random field. Thus, in a larger structure, the minimum of therandom material strength tends to be smaller. This randomness inevitably leads to the classical Weibull size effect (Fig. 4a) [12]. Toamalgamate the random and energetic triggers of failure, it is thus necessary to combine Eq. (11) and the classical Weibull size effectto form an energetic-statistical size effect equation for Type 1 failures [74,78]:

= ⎡⎣⎢

⎛⎝

⎞⎠

+ ⎤⎦⎥

∞σ f DD

rDDNc r

arn m

br/ 1/

(12)

where =Da constant, =m Weibull modulus, and =n number of spatial dimensions in which the failure mode and the structure arescaled. Since Eq. (12) accounts for both the energetic and statistical sources of size effect, it is often referred to as the energetic-statistical size effect law

Fig. 4b shows the comparison between Eq. (12) and the available experimental data on concrete beams from ten different labs[53,54]. It is worth noting that Eq. (12) was obtained by a rigorous combination (asymptotic matching) of fracture mechanics andstatistics [53,54]. An alternative derivation resting on a nano-mechanics based failure probability theory is given in [12,63].

Note that the number of free material parameters in the energetic law, Eq. (11), and energetic-statistical law, (12), is quite limited. Infact, the dimensionless energy release functions ′g (0) and ″g (0) can be calculated by finite element analysis, and so only cf needs to bedetermined to characterize Db. Accordingly, for cases in which the statistical effects are not significant, i.e., for the energetic model Eq.(11), the parameters are only three: cf (which can be correlated to the material strength and initial fracture energy by cohesive zonemodeling), r, and ∞fr . For cases in which the effects of material strength randomness are significant, the energetic-statistical model Eq.(12) can be used. In such a case the parameters are: c r D m, , ,f a and ∞fr . Note that Da is an additional parameter that describes thetransition from energetic to statistical size effect whereas m is the Weibull modulus, which controls the statistical size effect on themean behavior.

4. Description of boundary effect model

According to BEM, the size effect on structural strength is caused solely by the interaction between the FPZ and the boundary ofthe structure [31,35–37]. The BEM was derived in an empirical way, starting from the analysis of an edge crack in a semi-infiniteplate at incipient failure. In that case:

=K Y πa σIc Nc0 (13)

where =σNc nominal critical stress in the gross cross section, =Y 1.12 and =KIc material fracture toughness. Setting =σ fNc t andrearraning Eq. (13), Hu et al. defined the following transitional crack length:

⎜ ⎟ ⎜ ⎟= ⎛⎝

⎞⎠

≈ ⎛⎝

⎞⎠

∞∗a

πKf

Kf

11.12

14

Ic

t

Ic

t2

2 2

(14)

which is, of course, proportional to the well-known Irwin’s characteristic length [75], because ≈∞∗a l /4ch for plane stress and

≈ −∞∗a l ν/[4(1 )]ch

2 for plane strain are constants. For large-size structures, Hu et al. proposed that the failure condition should scaleas LEFM. Accordingly,

= =∞∗πa σ πa f K1.12 1.12Nc t c0 1 (15)

which leads to the following scaling law:

=∞∗

σf

a a/Nc

t

0 (16)

Then, by claiming that, for small sizes, →a 00 , the nominal strength should approach ft, Hu et al. postulated the following equationwithout any physical justification:

=+ ∞

∗σ

fa a1 /

Nct

0 (17)

which, unsurprisingly and conveniently, takes a similar form as the Type 2 SEL.Eq. (17) was derived by Hu et al. for very large plates weakened by an edge crack. For a finite size plate, they defined a different

nominal stress as the maximum stress in the net cross section, σn, which they suppose to account for the finiteness of the cross section.Without any justification, they assumed the stress distribution to be linear (Fig. 5a), and calculate σn by imposing equilibriumbetween the gross and net sections. Thus they obtain:

=σ σA α( )n

Nc

0 (18)

where =α a D/0 0 , and A α( ) is a dimensionless function dependent on the crack length and the structure geometry.The foregoing procedure leads to =∞

∗πa f Y α πa σ1.12 ( )t Nc0 0 , and so

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=∞∗

σ A αf

Y α a a( )

[ ( )/1.12] /n

t0

02

0 (19)

Finally,

=∞∗

σf

a a/n

t

e (20)

where = =a A α Y α a[ ( ) ( )/1.12]e 0 02

0 equivalent crack length accounting for geometry effects.Hu et al. then again conveniently assumed the following scaling relation to link the small- and large-size asymptotes:

=+ ∞

∗σ

fa a1 /

nt

e (21)

It is worthwhile to discuss the role of the nominal stress definition in the BEM (Eq. (21)). It is noted that the derivation of the Type 1and Type 2 SEL (Eqs. (11) and (5)) hinges on the energy release function, which gets automatically adjusted for different definitionsof σNc. Therefore, the final size effect equations are not affected by the definition of nominal strength. One may use any convenientchoice such as the average stress in the gross cross section [76,77,80–85]. However, this is not the case for the BEM. The BEM wasderived based on a particular definition of the nominal stress, which assumes a linear stress distribution along the ligament. As will beshown later, such a stress profile is just a fiction and differs grossly from the real stress distribution. This unrealistic assumptiondirectly enters the BEM equation, without any physical argument (Eq. (19)).

Attempting to correct various discrepancies, Hu et al. [37] introduced in the latest version of the BEM an additional artificialconcept, referred to as the “fictitious crack” (the same term as used by Hillerborg for a cohesive crack in his version of the cohesivecrack model for concrete). Their intent is to account, with the fictitious crack length, aΔ fic, for the effects of aggregate size;

=a βdΔ fic m, where dm, representing the maximum aggregate size, becomes an additional material parameter to be identified byfitting the experimental data. The stresses were assumed to be uniform along the initial fictitious crack and then to follow a lineardistribution in the remaining part of the ligament. Fig. 5b shows a schematic adapted from [37], describing the stresses in a typicalthree-point bending (3 PB) specimen. The relation between σn and the applied load Pmax is then found by imposing equilibrium. Basedon Fig. 5, the nominal stress for a 3 PB specimen can be calculated as follows:

=

=

=

+ − + − + − +

+ − + − + − +

+

σ P

P

P

n maxSL B

L βd L βd L βd L d L L βd β d

maxd ηl B

l β l β l β βl l βl β

maxη B

d l β

3 /

( ) ( ) ( ) 6 ( )

3 / /( ) ( ) ( ) 6 ( )

3 /2 ( 2 )

g T

g m g m g m g m g g m m

m T

Tm

2

3 4 2 2 2

3

3 4 2 2

(22)

where =S beam span, = ⩾ = − =l L d L D a/ 1,g m g 0 ligament length, =BT specimen thickness and =η S L/ g. Following Hu et al. [37],the peak load is then calculated combining Eqs. (21) and (22):

=+

∞∗

σ P βd l ηf

( , , , )1

n max mt

aa

e(23)

The foregoing equation describes the size effect on the nominal stress resulting from postulating the stress field in the ligament, σn.In the present contribution, to make a proper comparison with the experimental data and SEL, Eq. (23) is rewritten from Eq. (22)

in terms of the nominal stress defined in a usual way as =σ SP D B3 /2Nc max T2 :

Fig. 5. Stress distribution in the ligament assumed without justification in the (a) old version of BEM [34–36] and (b) latest version of BEM [37].

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=

=

= +

= +

−

−

+

+

∞∗

∞∗ ( )

σ

l β

β

( 2 )

1 2

NcP SD BP η α

DBf α

a adD

f A α

D α ad

D A α

323 (1 )

2(1 )

1 /

( )

1 ( / ) ( )

max

T

maxT

t

e

m

t

e

m

2

0

0

0

0 (24)

where = = =α a D a A α Y α α/ [ ( ) ( )/1.12]e e e 0 02

0 and = −A α α( ) (1 )0 02 for 3 PB specimens.

The foregoing BEM model depends on 3 parameters in total: the tensile strength ft, the mode I fracture toughness KIc and theparameter β that relates the size of the fictitious crack to the maximum aggregate size. According to Hu et al. [37], parameter β maychange with the structure size. However, the dependence of β on the structure size is not provided. Hu et al. propose using the medianamong the values of β that provide the best fit of the experimental data for different specimen sizes and different initial crack lengths.This kind of proposition precludes, of course, any predictive capability of the model. The value of β may lead to a satisfactory fit onlywithin the range of experimental data.

Finally, it should be stressed again that Eq. (24) has been obtained in an empirical way. Why does the effect of the equivalentcrack take the form + ∞

∗a a1/ 1 /e and not + ∞∗a a1/ 1/2 /e , or ∞

∗A a a/ /e ? Since Hu et al. obtained this expression empirically there isno answer to this question. In particular, the fact that Hu et al. did not verify the asymptotic behavior of this expression is a seriousdeficiency, as comparisons with the cohesive zone modeling and the bulk of experimental data will reveal in the sections that follow.

5. Deficiencies of BEM and comparisons to SEL

The limitations and inconsistencies of the previous versions of BEM were identified and comprehensively discussed in previoustwo papers [39,40]. The new version of BEM [37] not only persists in most of these deficiencies but also introduces new funda-mentally flawed concepts. This section will focus on the deficiencies of the new version whereas those of the older versions will bereviewed only briefly.

5.1. Application of BEM to Type 2 failures

The latest version of the Boundary Effect Model (BEM) [37] introduces the idea of using the maximum aggregate size dm to definethe stress profile (Fig. 5) and thus allows a more accurate prediction of the tensile strength. This idea, however, is fundamentallyflawed. Based on the stress profile depicted in Fig. 5, BEM considers the nominal strength σNc to be related to the strength σn as

= +σ σ βd D(1 / )Nc n a (25)

This scaling relation, which is the consequence of introducing a fixed length scale dm into the stress profile, is similar to the boundarylayer model for the size effect on the modulus of rupture [86]. By further imposing a size effect on σn, Eq. (21), the latest version of theBEM, Eq. (24), yields the scaling relation for the nominal strength, σNc:

∝ + +∞∗ −σ ζD a βd D(1 / ) (1 / )Nc m

1/2 (26)

This scaling equation predicts that: (1) ∝ −σ DNc1 at the small-size limit ( →D 0), and (2) ∝ −σ DNc

1/2 at the large size limit ( → ∞D ).But the small-size asymptotic behavior is blatantly unrealistic. It violates the second-order asymptotic properties of the cohesive crackmodel. As will be shown in Section 5.1, in contrast to SEL, the deviation of BEM from cohesive fracture mechanics occurs well beforethe structure size D gets close to the aggregate size.

This deficiency can be demonstrated by comparing SEL and BEM to the size effect curves obtained by means of the cohesive zonemodel. Assuming, for simplicity, a linear cohesive law, and following Cusatis and Schauffert [61], it is convenient to rewrite Eq. (24)in the form =ξ F X( ) with = ′ξ g f σ1/ ( / )t N0

2 and = ′X g D g l/ ch0 0 . Assuming further a plane strain condition, one can find the followingrelationship between the dimensionless energy release rate g α( )0 and the geometry function Y α( )0 :

= −g α πα ν Y α( ) (1 ) ( )0 02

02 (27)

Then, starting from Eq. (24) and rearranging, one obtains:

⎜ ⎟⎜ ⎟=′

⎛⎝

⎞⎠

= ⎛

⎝ ′+ ⎞

⎠

⎛

⎝⎜ +

′⎞

⎠⎟

−

ξg

fσ A α g

X βg d l

Xg A α1 1

( )1 2

/( )

t

N

m ch

0

2

02

0

0

0 0

2

(28)

On the other hand, considering that, according to [61], ≈ ∗c E G f0.44 /f f t2, Type 2 SEL, Eq. (4), can be written as follows:

= +ξ X0.44 (29)

While both BEM and SEL predict a linear relation between ξ and X for large sizes, the intercept in BEM depends significantly onthe particular structural geometry under consideration, the relative crack length α0 and the maximum aggregate size dm. This de-pendence disagrees with cohesive fracture mechanics. Cusatis et al. [61] showed that, for a linear cohesive law, F X( ) tendsasymptotically to SEL with an intercept equal to 0.44 regardless of the structure or specimen geometry, and of the relative cracklength.

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To exemplify the problem, let us compare the size effect curves obtained by cohesive fracture mechanics for a linear cohesive lawto the ones predicted by SEL and BEM, considering geometrically-scaled three-point bending (3 PB) specimens. In that case, for

=S D/ 4, according to Anderson [85],

= −

= − − − ++ −

A α α

Y α

( ) (1 )

( ) α α α απ α α

2

1.99 (1 )(2.15 3.93 2.7 )(1 2 )(1 )

2

3/2

The foregoing expressions can be used to calculate the size effect curve predicted by BEM, Eq. (28). For the cohesive crack model, thesize effect curve has been calculated from the following expression proposed by Cusatis [61]:

= + +− ′ +

ξ X Xα g X

1 119(1 ) 254

0 (30)

Fig. 6a–d show a comparison between SEL, BEM and the Cohesive Size Effect Curves (CSEC) for two dimensionless crack lengths,namely =α 0.150 and 0.3. For the calculations, the following material properties are used: =E 40 GPa and =ν 0.18 (typical elasticproperties of concrete), =f 5.56t MPa, =K 1.49Ic MPa m1/2 and =d 10m mm (taken from [37]). It is interesting that while the onlyparameters required for SEL and CSEC are tensile strength ft and fracture energy Gf , BEM requires an additional model parameter βrelated to the maximum aggregate size (instead of just the basic two influencing parameters, ft and Gf ). Yet, notwithstanding thepresence of a third fitting parameter, both the small and large-scale asymptotes predicted by BEM are nowhere close to cohesivefracture mechanics.

For =β 0, corresponding to the old version of BEM [34–36], the size effect predicted by BEM is described by a straight line of unitslope, similar to Type 2 SEL. However, there is a substantial difference: while the CSEC curve always closely agrees with the SEL forsmall sizes and tends asymptotically to the SEL for sufficiently large sizes, the same is not true for the BEM, which is generally

Fig. 6. Comparison between Size Effect Law (SEL), Boundary Effect Model (BEM) and Cohesive Size Effect Curve (CSEC) for geometrically-scaledthree-point bending specimens and different initial notch lengths: (a) case =α 0.150 and (b) related magnification for lower X values; (c) case

=α 0.30 and (b) related magnification for lower X values. The following material properties are used: =E 40 GPa, =ν 0.18 (typical elastic propertiesof concrete), =f 5.56t MPa and =K 1.49Ic MPa m1/2 and =d 10m mm (data taken from [37]).

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characterized by a higher intercept, depending on the initial notch length and the specimen geometry through the function A α( )0 .The disagreement, especially in terms of the tangent of the size effect curves, gets worse for values of β other than zero, as

proposed in the latest formulation of the BEM [37]. In this case, the small-scale behavior totally differs from CSEC, and there is nopossible β-value that could mitigate it. This is not surprising because the BEM introduced an artificial size effect term + βd D(1 / )a (Eq.(25)), which is based on an unrealistic stress distribution. The validity of such a term could easily be checked by asymptotic analysis,which Hu et al. skipped. On the other hand, the SEL shows an excellent agreement, which reflects the physical soundness of themodel.

Finally, it is important to note that, aside from conflicting with cohesive fracture mechanics, the asymptotic behavior of BEMdisagrees with the bulk of experimental data on quasibrittle materials, as shown next.

5.2. Application of BEM to Type 1 failures

In [37], Hu et al. claimed that the BEM can be applied to Type 1 failures. In the old version of BEM [34–36], Hu et al. imposed forthis purpose an initial crack length, aeff , that is proportional to the specimen size, claiming to capture the Weibull size effect. But thisclaim is groundless. The Weibull size effect applies to quasibrittle structures only if they are very large (i.e., much larger than theinhomogeneity size) and only if they fail at macrocrack initiation. The key statistical characteristic of the Weibull size effect is theWeibull modulus. But the BEM involves no statistical characteristics. Therefore, it cannot capture the Weibull size effect.

The proportionality between the flaw size and specimen size, assumed in [34–36], was also incorrect. Freudenthal showed thatthe Weibull size effect can be correlated to flaw statistics, for which he assumed the Fréchet distribution of the largest flaw sizes. It is,of course, reasonable to state that for a larger specimen the maximum flaw size will be larger, more critical. However, based on theextreme value statistics (Fréchet distribution), the mean value of the largest flaw size increases only mildly with the specimen size.Assuming the flaw size to be proportional to the specimen size, the BEM would predict the large-size asymptotic scaling of −D 1/2. Suchscaling would, of course, be incorrect, and is never seen in Type 1 size effect tests. The size effect data on three-point and four-pointbend tests on concrete materials have consistently shown that the large-size asymptote of the two-dimensional size effect is −D 1/12

[12,53,54,74,78] which is based on the known Weibull modulus for concrete, ≈m 24 (see e.g. Fig. 4).As mentioned earlier, in the newest version of BEM [37], Hu et al. tried to circumvent the problem by introducing an additional

fictitious crack proportional to the maximum aggregate size, =a βdΔ fic m (see Fig. 5b). The additional fictitious crack is supposed torepresent the initial micro-damage that is present even without a stress-free crack or notch, and the stresses along this crack areassumed to be uniform (which is certainly far from reality).

In contrast to the old BEM, the fictitious crack in the new BEM does not influence the definition of the equivalent crack ae, whichis set to 0 in the case of failure at crack initiation. Rather, it influences the definition of the nominal strength σn as described in Eq.(22). Setting =α 00 and =a 0e in Eq. (24), which describes the size effect on the nominal strength in the gross section according toBEM, one gets:

= +σ f βd D(1 2 / )Nc t m (31)

It is interesting that this new expression, which was described by Hu et al. as a substantial improvement, resembles the earlyversion of size effect equation on the modulus of rupture [86] (special case of Eq. (12) with =r 1). However, it has been shown thatthe optimum fitting of test data often requires some different values of r. Meanwhile, Eq. (31) predicted an incorrect large-sizeasymptotic behavior, i.e. → =σ f const.Nc t . This again contradicts all experimental evidence, as can be noted from, e.g., Fig. 7a and b

Fig. 7. Fitting of experimental data on geometrically-scaled un-notched specimens made of asphalt concrete by Type 1 Size Effect Law (SEL) andBoundary Effect Model (BEM). Data taken from [64]. The strength is calculated averaging 12 test results for size 1, 28 for size 2, 30 for size 3 and 7for size 4 [(a) shows the effect of parameter β on BEM]. The strength is fixed at 12.4 MPa, which is the average of 28 tests on size 2 specimens; (b)Best fitting with BEM and Type 1 SEL. A Weibull modulus =m 26, as obtained from independent histogram testing, was adopted for Type 1 - SEL.

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reporting the test data taken from [64] on geometrically-scaled, notch-free three-point-bending (3 PB) specimens and uniaxial tensionspecimens made of asphalt mixtures. In these experiments, the size of the uniaxial tension specimen was chosen to be much largerthan the aggregate size. Therefore, one may use the classical Weibull theory to determine the mean strength of the tensile specimen.Meanwhile, one can also calculate the mean strength of a 3 PB specimen by the Weibull theory. The equivalent size of the 3 PBspecimen that has the same mean strength as the tensile specimen can then be calculated [50,64]. Consequently, the size range inthese tests is sufficiently large to show a clear statistical size effect. For the two intermediate sizes of 3 PB beams, histogram testingwas performed so that the Weibull modulus of the material can be determined ( =m 26). As expected, this modulus is reflected in theslope of the large-size asymptote of the average strength as can be noted in Fig. 7a and b.

The experimental data are complemented by the optimum fitting of the finite weakest-link model [12,60,62–64] which caninherently capture the transition from energetic to statistical size effect. As can be noted (Fig. 7a and b), the Type 1 SEL provides anexcellent fitting of the data, particularly at the small- and large-size asymptotes. Satisfying the correct asymptotic behaviors is anessential requirement of any physical model. It is important to note that the Type 1 SEL satisfies this requirement, by exploitingparameters such as the Weibull modulus, m, which can be verified independently by other tests, e.g., by strength histogram testing.

Owing to the empiricism of its formulation, the same cannot be said for BEM, which neglects the statistical size effect for largesizes. This is clear in Fig. 7a, which shows the influence of parameter β on the BEM size effect. The strength for these analyzes was setequal to 12.4 MPa, which is the average of 28 tests on size 2 specimens. This is the intermediate size of all the specimens investigated.Any other strength value could have been used to investigate the effects of β without changing the conclusions. For =β 0, no sizeeffect is predicted, which contradicts experimental data. For increasing values of β, the small-scale asymptote proportional to −D 1

extends further and further.For some β-values, the BEM is able to fit some of the small-size data well, but not for large sizes. This is evident in Fig. 7b, which

presents the best fit possible of BEM obtained by means of the Levemberg-Marquardt algorithm [87,88]. The underlying reason is thatthe BEM ignores the statistical size effect, which governs the large-size asymptote. This is a serious issue because the BEM will grosslyoverestimate the mean strength of large-size structures, which would lead to an unsafe design.

In view of these observations, one might wonder: How, in the latest Hu et al.-s paper [37], could the BEM (Eq. (31)) give anacceptable fit of the experimental data on Type 1 size effect? Obviously, because the size range of the test data fitted was ratherlimited and the asymptotic behavior, dominated by the statistical effects, was ignored.

5.3. Limited practical applicability

As discussed in the foregoing sections, the application of BEM to finite-size structures relies on an assumed stress distributionalong the ligament. In fact, the σn used in the BEM represents the maximum stress at the crack tip assuming a linear distribution ofstresses in the net section (see Fig. 5a and b). In contrast, the nominal strength =σ c P A/N s is a generic measure of the load P with adimension of stress. Here cs = conveniently chosen dimensionless constant, used to make σN equal to the stress at a certain chosenpoint (for beam specimens, we choose =c L D3 /2s where L = beam length).

In standard Mode I fracture specimens, and only if the specimen geometry, including a D/ , does not change, σn and σN are mutuallyproportional and thus the size effects are the same.

However, for geometries other than simple Mode I fracture specimens, the definition of the stress profile is usually ambiguous. Infact, the fracture process may occur under mixed mode conditions, or the direction angle ϕ in which the ligament will crack may be apriori unclear (Fig. 8a–c). Further, the ratio of bending moment M to normal force N in the ligament may often be statically in-determinate (Fig. 8d,f) and, in some cases, the stress profile cannot even be defined (Fig. 8e–h). For all the foregoing situations, thefunction A α( )0 cannot be calculated. Hence, BEM cannot be applied.

By contrast, the size effect law does not rely on knowing the direction of crack propagation. Rather, it relies on the energy releaserate at the crack tip. The dimensionless energy release rate function g α( )0 used in SEL can be unambiguously and easily calculated forall these cases. For example, despite the complex shape of the dominant cracks, the SEL is applicable to the shear failure of reinforcedconcrete beams and to punching shear failure of slabs (and, for both, is included in the 2019 version of ACI Building Code, ACIStandard 318).

5.4. Groundless assumption of stress profile in the ligament

It has been pointed out that the BEM strongly relies on the definition of the stress profile in the ligament, which directly affects thesize effect equation. Therefore, it is important to examine the correctness of the assumed stress profile. The latest version of the BEM[37] assumes a uniform stress profile within a portion of the ligament whose length is proportional to the maximum aggregate size,and a linear stress profile through the rest of the ligament. This assumption is in conflict with the cohesive zone model simulations forany structure size, as can be noted from Fig. 9a–c that compare the stress distributions at peak load postulated by BEM to thosepredicted by a cohesive zone model featuring a simple linear softening cohesive law.

The simulations were done for 3 PB specimens with span-depth ratio = =S D E/ 4, 40 GPa, =ν 0.18 (typical elastic properties ofconcrete), =f 5.56t MPa, and =K 1.49Ic MPa m1/2 (taken from [37]). The cohesive law was implemented in a FE model in ABAQUS/Explicit [89]. The models combined 4-node bi-linear plain-strain quadrilateral elements (CPE4R) with a linear elastic isotropicbehavior and 4-node two-dimensional cohesive elements (COH2D4) with a linear traction-separation law to model the crack. ForBEM, the maximum aggregate size =d 10m mm was used. Three cases were considered: (a) ligament length ≈L l6g ch, correspondingto a typical medium size specimen, (b) →L lg ch, and (c) →L dg m.

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When the specimen size is finite but relatively large compared to Irwin’s characteristic length, Fig. 9a, the FPZ may not be fullydeveloped, as shown by the normalized plot of the predicted cohesive stresses. For this case, the cohesive stresses take a nonzerovalue at the crack tip and increase monotonically up to the point of tensile strength limit, which advances through the ligament.Beyond that point, the stress decreases monotonically and turns to compression in the upper part of the ligament. The stress dis-tribution postulated in BEM is nowhere close this picture, regardless of the β value. This is particularly evident if one considers thecases in Fig. 9b and c. When the ligament is close to lch, as shown in Fig. 9b, the crack openings at peak load are significantly smallercompared to the previous case.

Fig. 8. Examples of specimens and structures for which SEL can be used and BEM cannot.

Fig. 9. Stress distribution along the ligament of a 3 PB specimen contrasting BEM versus a cohesive zone model featuring a linear cohesive law: (a)≈L l6g ch, (b) →L lg ch and (c) →L dg m. For all the simulations = =S D E/ 4, 40 GPa, =ν 0.18 (typical elastic properties of concrete), =f 5.56t MPa

and =K 1.49Ic MPa m1/2 (taken from [37]).

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Hence, the stress at the crack tip is nonzero and closer to the material strength. Moving away from the tip, the stress increases upto strength limit and then decreases abruptly in the remainder of the ligament. Since the relative distance from the tip to reach thetensile strength point is larger than it is in the previous case, the equilibrium requires a higher negative slope. These aspects, ofcourse, cannot be captured by BEM.

For the case of =β 0 (which corresponds to the old version of BEM [34–36]), the slope of the linear stress distribution must beconstant. For =β 1, the approximation is slightly improved in the second part of the ligament but the approximation of the stresses inthe FPZ remains still very coarse. For =β 4, the initial overestimation of the stresses caused by the assumption of uniform distributionleads to a significant overestimation of the negative slope of the ligament portion described by a linear distribution. Similar con-clusions can be drawn for the case →L dg m, Fig. 9c, for which BEM predicts a uniform stress distribution equal to ft followed by alinear stress distribution with an infinite negative slope.

The foregoing observations are also supported by recent experimental analyzes obtained by Digital Image Correlation (DIC) whichcompletely disprove the stress distribution assumed in BEM [52]. Dr. Carloni, in collaboration with Dr. Maria Chiara Bignozzi fromUniversity of Bologna, have recently conducted a pilot study on Alkali Activated Mortars (AAM). The fracture properties of themortars have been determined from three-point bend tests of notched beams following a draft of the report prepared by ACI 446Committee (Fig. 10a). The dimensions of the notched beams were 70mm (width B) −70mm (depth D) 300mm (length L). The netspan S was equal to 3D. The notch a0 was D/3. Digital image correlation was used to determine the strain profile along the ligament.

An example of the strain profile is reported in Fig. 10b, which refers to an AAM with coal Fly Ash (FA) as precursor for thesynthesis. FA was characterized by a low content of calcium and iron oxides, while nearly 80 wt% was made of silicon and aluminumoxides. As alkaline activators, 8M sodium hydroxide and sodium silicate solutions were used. Fine silica sand with maximum ag-gregate size =d 2a mm was used. The strain profile along the ligament was obtained by averaging εxx over a 5mm strip centered withrespect to the notch. The strain profiles of Fig. 10b correspond to different points of the load-load point displacement ( −P δ) curvereported in Fig. 10c. A vertical dashed line indicates the tensile strain εt sp, corresponding to the splitting tensile strength measured atthe age of testing. The intersection between the strain profile and the dashed line corresponding to εt sp, is indicated with a marker forpoints C, D and E of Fig. 10c. It can be observed that the neutral axis moves upward as the Fracture Process Zone (FPZ) develops. Atpoint C, which corresponds to the peak load, the distance from the notch tip to the marker that indicates the beginning of the FPZ canbe estimated to be d2 a.

5.5. Lack of mathematical basis for asymptotic matching

Since the procedure to set up BEM was intuitive and empirical, it is no surprise that no use was made of any systematic matchingof the two-sided second-order asymptotic properties characterizing the way the asymptotes are approached. In the spirit of thisintuitive procedure, there is no reason for not using, e.g., = + ∞

∗σ f a a/ 1/2 /n t e , or any other variant with the same asymptotesinstead of Eq. (21). Probably the tacit reason for choosing Eq. (21) was to arrive at an expression similar, though not identical, toType 2 SEL, because the SEL has become, for quasibrittle materials, a generally accepted theoretical result with solid experimentalsupport.

It should be emphasized, though, that the SEL, unlike BEM, was derived by matching the second-order asymptotic properties ofthe cohesive crack model [39,61] or the equivalent LEFM [1]. This difference is conspicuous in the comparisons of SEL, BEM and thecohesive zone model presented in the previous sections.

5.6. Comparison with experimental data and comments on Hu’s optimum data fitting

Fig. 11a and b compare the analysis of Hoover et al.’s tests on geometrically-scaled, notched concrete beams [41,42] by SEL andby the latest version of the BEM, for two different initial notch lengths: =α 0.150 and =α 0.30 . To asses the predictive capability of thetwo models, both SEL and BEM were calibrated by best fitting of the experimental data for the notch length =α 0.150 (Fig. 11a). Thefitting was based on all the experimental data, in order to consider the scatter. Then, the calibrated set of parameters was used to

Fig. 10. Experimental characterization of the strain in the ligament of three-point-bending specimens made of Alkali Activated Mortar (AAM): (a)test setup; (b) strain profile at different loading conditions; (c) load-diplacement curve and corresponding loading conditions.

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predict the structural strength for the case =α 0.30 (Fig. 11b).The optimal fitting by SEL resulted into an estimated fracture toughness =K 1.49IC MPa m1/2 and an effective crack =c 21.0f mm

[42]. Exploiting the link between cf and material strength proposed by Cusatis and Schauffert [61] and the subsequent refinementproposed by Yu et al. [39], the material tensile strength calculated from SEL is = =f K c0.29/ 5.23t Ic f MPa. Fig. 11b shows the sizeeffect predicted by SEL leveraging the foregoing results and recalculating the dimensionless energy release functions g α( ) and ′g α( )for =α 0.30 . The CoV of errors of fit (normalized by data mean) provided by the best fit through SEL is 4.15% and 2.31% for =α 0.150and = =α a D/ 0.300 0 , respectively.

As seen in the figures, not only the Type 2 SEL is in excellent agreement with the experimental data but also the asymptoticbehavior for both small and large sizes follows the existing, extensively validated, physical models. For large sizes, the SEL convergesasymptotically to the LEFM whereas, for small enough sizes, the structural strength tends to the strength limit. It is important to stresshere again that the curve shown in Fig. 11b, which is in excellent agreement with the experimental data, is a prediction made by theSEL, with the main parameters calibrated against another set of data.

The BEM, in contrast, is characterized by a completely different asymptotic behavior, especially for small beam sizes. This is veryclear from Fig. 11a and b each featuring the following cases: (1) BEM – 1: best fitting obtained by jointly optimizing K f,Ic t and β, and(2) BEM – 2: best fitting obtained by fixing =K 1.49Ic MPa m1/2 and =f 5.59t MPa according to [37] and optimizing β. As can benoted, for both =α 0.150 (best fitting) and =α 0.300 (prediction), BEM provides an excessive, unrealistically high, structural strength,which increases (at the unreasonable slope of D1/ ) in the small size range.

A consequence of the wrong asymptotics in BEM is that its predictions are strongly affected by the value chosen for the parameterβ. To give an example, the best fits shown in Fig. 11a provided = =β K0.11, 1.61Ic MPa m1/2 =f 7.40t MPa with a CoV of 0.91% and

= =β K0.61, 1.49Ic MPa m1/2 =f 5.59t MPa with a CoV of 7.93%, respectively. However, with the same fitting parameters, theprediction for the case =α 0.300 was rather inaccurate. Both fittings exhibited a CoV of 50.24% and 45.69 %, respectively, despite thelimited data range.

Of course, the agreement between BEM and the experimental data can be improved by including the data for all the notch lengthsin the calibration of the parameters, as proposed in [37]. However, the fact that a decent agreement could be achieved in such a casedoes not imply a predictive ability of the BEM. In contrast, the very poor prediction obtained by calibrating the BEM using a subset ofthe experimental data (Fig. 11a and b) inevitably shows the importance of capturing the asymptotic behavior and gives anotherreason for doubting the applicability of BEM in structural design. Further comparisons between SEL and BEM, with related dis-cussions, can be found in [39,40].

6. Conclusions

1. The BEM is not based on the cohesive crack model and violates its asymptotic properties. The structure of the BEM does not evenallow asymptotic matching of cohesive crack model properties.

2. Although the maximum aggregate size, dm, affects the properties of the cohesive crack model, the intuitive way by which theeffect of dm is introduced in the latest version of the BEM gives an incorrect small-size asymptote of the size effect curve.

3. Because of the ambiguity in defining the required stress profile over the ligament, the BEM cannot be applied to specimens orstructures other than the standard mode I fracture specimens.

Fig. 11. Optimum fit of Hoover et al. test data [41,42] by SEL and BEM for (a) =α 0.150 and (b) =α 0.30 .

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4. The stress distribution across the ligament is not calculated but assumed, and conflicts with what is obtained from the cohesivecrack model.

5. Comparisons with the test data on size effect reveal extensive disagreements. Especially, the optimum fit by the BEM grosslydeviates from the results of most extensive series of tests conducted so far—Hoover et al.’s tests of notched beams with a verybroad range of size and notch depths. In particular, it is shown that when BEM is used as a predictive theory, the CoV of errors offit can be significant, in the order of 50%. In contrast, the predictions of SEL are always in excellent agreement, with a CoV in theorder of 4 %.

6. The BEM does not properly distinguish between Type 1 and 2 size effects.7. Since it does not include any description of the randomness of the material strength, the BEM cannot capture the statistical size

effect.8. Although the proximity of a boundary does influence the fracture behavior, this influence is more complex than considered in the

BEM and cannot be reduced to a simple principle. But the cohesive crack band model with a finite RVE automatically capturesthis influence.

9. Because of the aforementioned problems, the BEM, unlike the SEL, is not applicable to structural design, is not relevant todeveloping the design code provisions for size effects on structural strength, cannot be used as the basis for measuring thefracture energy, and is not transplantable to other quasibrittle materials.

10. The foregoing conclusions and the BEM critique are important for all quasibrittle materials, especially aircraft fiber composites,tough ceramics, sea ice, large landslides, rock, stiff soils, micrometer-scale devices, bone and other biomaterials, and biomi-metics, for all of which the non-statistical size effect has yet to become properly recognized in design and engineering practice.

Acknowledgments

CC would like to acknowledge the start up fund provided by Case Western Reserve University, which was partially used to supportthis research. MS acknowledges the financial support from the Haythornthwaite Foundation through the ASME HaythornthwaiteYoung Investigator Award. JL thanks NSF for partial support under grant CMMI- 1361868 to the University of Minnesota. CGH thanksArizona State University and the Ira A. Fulton Schools of Engineering for providing research startup funds to conduct this research.ZPB thanks ARO for partial support under grant W911NF-19-1-0039 to Northwestern University. Discussions in ACI Committee 446,Fracture Mechanics, under the current chair CC and past chair CG, were particularly helpful.

Appendix A. General derivation of SEL from energy conservation and dimensional analysis

The release of complementary strain energy Π engendered by fracture is a function of: (1) the length a of the fracture (or of crackband) at maximum load, and (2) the area of the zone damaged by fracturing, which is w ac , where =w ndc m = material con-stant=width of crack band swept by fracture process zone width during propagation of the main crack, dm = maximum aggregatesize, and n = 2 to 3. According to Buckingham theorem of dimensional analysis, Π must be expressed as a function of dimensionlessparameters, which may be taken as =α a D/1 and =α w a D/c2

2. Then

= ⎛⎝

⎞⎠E

PbD

bD f α αΠ 12

( , )2

21 2 (32)

For geometrically similar structures of different sizes, f must be a smooth function independent of D. The energy conservation duringcrack growth necessitates that ∂ ∂ =a G bΠ/ f , where Gf = critical value of energy release rate. Noting that

∂∂

=∂∂

∂∂

+∂∂

∂∂

fa

fα

αa

fα

αa1

1

2

2

(33)

where ∂ ∂ =α a D/ 1/1 and ∂ ∂ =α a w D/ /c22, we consider the first two linear terms of a Taylor series expansion,

≈ + +f α α f f α f α( , ) (0, 0)1 2 1 1 2 2, where = ∂ ∂ = ∂ ∂f f α f f α/ , /1 1 2 2. Setting =P bdσ , we obtain

⎜ ⎟⎛⎝

+ ⎞⎠

=fD

f wD

PbE

G b2

cf

1 22

2

(34)

Rearranging, setting =P bDσNc, and solving the foregoing equation for σNc (nominal strength of structure), we obtain from Eq. (34)the deterministic (or energetic) size effect in Eq. (5), in which =D w f f/c0 2 1 = constant (independent of size D), and =Bf G E f w2 /t f c2 .Note that this derivation rests only on the energy conservation principle and on the hypothesis that the size, wc (width or length), ofthe fracture process zone at the front of dominant crack is a constant (a material property). Only geometrically similar structures areconsidered, and a is assumed to be non-zero, which means Type 2 size effect.

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